Syzygy gap fractals--I. Some structural results and an upper bound

TL;DR

This paper studies syzygy gap fractals associated with linear forms over fields of positive characteristic, revealing their structure, self-similarity, and providing an upper bound for their local maxima, with applications to Hilbert--Kunz theory.

Contribution

It establishes the structural properties of syzygy gap fractals, including their determination by zeros and behavior near maxima, and proves an upper bound conjectured by Monsky.

Findings

Syzygy gap fractals are determined by their zeros.

They exhibit self-similarity related to 'magnification by p'.

An upper bound for their local maxima is derived.

Abstract

k is a field of characteristic p>0, and l_1,...,l_n are linear forms in k[x,y]. Intending applications to Hilbert--Kunz theory, to each triple C=(F,G,H) of nonzero homogeneous elements of k[x,y] we associate a function delta_C that encodes the "syzygy gaps" of F^q, G^q, and H^q*l_1^{a_1}*...*l_n^{a_n}, for all q=p^e and a_i<= q. These are close relatives of functions introduced in "p-Fractals and power series--I" [P. Monsky, P. Teixeira, p-Fractals and power series--I. Some 2 variable results, J. Algebra 280 (2004) 505--536]. Like their relatives, the delta_C exhibit surprising self-similarity related to "magnification by p," and knowledge of their structure allows the explicit computation of various Hilbert--Kunz functions. We show that these "syzygy gap fractals" are determined by their zeros and have a simple behavior near their local maxima, and derive an upper bound for their local maxima which has long been conjectured by Monsky. Our results will allow us, in a sequel to this paper, to determine the structure of the delta_C by studying the vanishing of certain determinants.